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Rumely R.S. Capacity Theory on Algebraic Curves
Rumely R.S.  Capacity Theory on Algebraic Curves









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: Capacity Theory on Algebraic Curves

: Rumely R.S.

: en

: /

:

ed2k: ed2k stats

: 1989

: 437

: 14.05.2008

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$A_{\zeta}$--capacitability      259
$A_{\zeta}$--capacitability, pathological examples      264266
$CPA(\mathcal{R}(\mathcal{C})_v)$      113 115
$e(\tilde{K}_A)$      322 336
$E^*_{\zeta}$      140 146 195
$Gal(\tilde{K}/K)$, action on probability vectors      321
$i_v(x, y)$ for curves of arbitrary genus      96 107
$i_v(x, y)$ for Tate curves      9496
$i_{\zeta}(x, y)$      96 107
$j_v(x,y)$ for curves of any genus      116
$j_v(x,y)$ for Tate curves      9496
$j_z(x, y)$, $j_{\zeta}(x, y)$      107 109 110 222
$j_z(x, y)$, $j_{\zeta}(x, y)$, continuity of      112
$j_z(x, y)$, $j_{\zeta}(x, y)$, extension to $\mathcal{C}(\bar{K}_v)$      120
$j_z(x, y)$, $j_{\zeta}(x, y)$, mean value property for      108
$j_z(x, y)$, $j_{\zeta}(x, y)$, polarization identity for      112
$j_z(x, y)$, $j_{\zeta}(x, y)$, satisfies Laplace equation      109
$j_z(x, y)$, $j_{\zeta}(x, y)$, takes rational values at rational points of $\mathcal{R}(\mathcal{C})_v$      112
$K_v$-symmetric set $\mathbb{F} \in \mathcal{C}(\Omega_v)$      311 321
$PL_{\zeta}$-domain, definition      50 236
$PL_{\zeta}$-domain, finite unions of isometrically parametrizable balls      252
$PL_{\zeta}$-domain, independent of $\zeta$      244 252
$q_v$      20
$S_0(F,\zeta)$      292 293
$S_{\alpha}(\{E_j\})$, $S_{\alpha}(\{D_j\})$      285
$u_{\nu}(z,\zeta)$      135
$V_{\zeta}(E)$ (Robbins constant), bound for unions of sets      148 198
$V_{\zeta}(E)$ (Robbins constant), definition      136 190
$V_{\zeta}(E)$ (Robbins constant), extremal properties      136 147 197
$\Delta_{\chi}$ (sum of slopes operator)      113 256
$\log_v(x)$      20 185
$\mathcal{R}$-disc      222
$\mathfrak{X}$-capacitable set      327 414
$\mathfrak{X}$-trivial set      323
$\mathfrak{X}, s)$-function      383
$\partial D_{\zeta}$. (outer boundary)      143 146
$\tilde{K}_A$      336
A-disc, A-codisc      224
A.C. sets are approximatable by PL-domains, RL-domains      311316
A.C. sets are closed      276
Abel map      48
Abelian integral      64
Abhyankar, S.      97
Absolute values (conventions)      20
Adelic neighborhood      322 336
Adelic set      322 336
Ahlfors, L.      160 162
Algebraic capacitability      15 258
Algebraic capacitability, Choquets theorem fails      265
Algebraic capacitability, compact sets are A.C.      262
Algebraic capacitability, finite unions of open balls, closed balls are A.C.      274
Algebraic capacitability, finite unions, intersections of RL-domains, compact sets are A.C.      266
Algebraic capacitability, non-A.C. sets      263264
Algebraic capacitability, not stable under intersections      266
Algebraic capacitability, PL-domains are A.C.      258
Algebraic capacitability, pullbacks of A.C. sets are A.C.      274 304309
Algebraic capacitability, RL-domains are A.C.      260
Algebraic curves (conventions)      21
Algebraic integers      1 9 373 376
Algorithm for finding $val(\Gamma)$      367
Algorithm for finding minimal model      101
Analytic arc      162 165 167
Approximation of $G(z, \zeta;E)$, (Archimedean case), basic theorem      168
Approximation of $G(z, \zeta;E)$, (Archimedean case), independent variability of leading coeffs      173184
Approximation of $G(z, \zeta;E)$, (Archimedean case), input to Fekete Szego theorem      171 173
Approximation of $G(z, \zeta;E)$, (Nonarchimedean case), input to Fekete Szego theorem      318 319
Arakelov function $(x, y)_v$      13 14 80 89
Arakelov function $(x, y)_v$, (Archimedean case) for $P^1$      80
Arakelov function $(x, y)_v$, (Archimedean case) for curves of genus $g \geq 2$      83
Arakelov function $(x, y)_v$, (Archimedean case) for elliptic curves      82
Arakelov function $(x, y)_v$, (Archimedean case), axioms for Arakelov Greens functions      80
Arakelov function $(x, y)_v$, (Archimedean case), in terms of $[x, y]_{\zeta}$      86
Arakelov function $(x, y)_v$, (Archimedean case), non-normalized      80
Arakelov function $(x, y)_v$, (Nonarchimedean case) for $P^1$      90
Arakelov function $(x, y)_v$, (Nonarchimedean case) for curves of genus $g \geq 2$      96 115116
Arakelov function $(x, y)_v$, (Nonarchimedean case) for curves w/good reduction      90
Arakelov function $(x, y)_v$, (Nonarchimedean case) for Tate curves      93
Arakelov function $(x, y)_v$, (Nonarchimedean case), comparability with $\|x, y\|_v$      128
Arakelov function $(x, y)_v$, (Nonarchimedean case), definition      89
Artin contraction theorem      19
Artin, M.      97 118
Balancing      397
Ball, boundary of      30
Ball, conventions      22 30 185
Ball, isometrically parametrizable      31 185186
Ball, parametrizable      30
Ball, rational, irrational      186
Barrier      143
Basis functions      382
Bent line segments      342
Bertrandias, F.      2
Blowing up      101 102 117 118 121
Borel measure      135 186
Boundary of ball or disc      30
Boundary of disc or codisc      239
Boundary of island domain $\nabla D$      239
Boundary of R-disc      239
Boundary of RL-domain      50
Bourbaki, H.      45 321
Brown, S.      30
Bsilinson, A.A.      18
Canonical distance $[x, y]_{\zeta}$      12 14
Canonical distance $[x, y]_{\zeta}$, (Nonarchimedean case) for $P^1$      90
Canonical distance $[x, y]_{\zeta}$, (Nonarchimedean case) for curves w/good reduction      90
Canonical distance $[x, y]_{\zeta}$, (Nonarchimedean case) for Tate curves      93
Canonical distance $[x, y]_{\zeta}$, (Nonarchimedean case) local ultrametric inequality      130
Canonical distance $[x, y]_{\zeta}$, (Nonarchimedean case), weak triangle inequality      127
Canonical distance $[x, y]_{\zeta}$, approximatable by $(1/N)\log|f(z)|$      72
Canonical distance $[x, y]_{\zeta}$, change of centers formula      72
Canonical distance $[x, y]_{\zeta}$, characterization of      69
Canonical distance $[x, y]_{\zeta}$, construction of      58 63
Canonical distance $[x, y]_{\zeta}$, definition      57
Canonical distance $[x, y]_{\zeta}$, factorization property      57
Canonical distance $[x, y]_{\zeta}$, Galois stability      57 61
Canonical distance $[x, y]_{\zeta}$, Grosss formula for      13 77 106
Canonical distance $[x, y]_{\zeta}$, in terms of Nerons pairing      77 106
Canonical distance $[x, y]_{\zeta}$, invariance under base change      90
Canonical distance $[x, y]_{\zeta}$, joint continuity in x, y      57 61 63 66 67 6970
Canonical distance $[x, y]_{\zeta}$, weak triangle inequality      73 127
Canonical metric $\|x, y\|_{\mathfrak{C},v}$      103 221
Canonical metric $\|x, y\|_{\mathfrak{C},v}$, comparability with $\|x, y\|_v$      120
Canonical metric $\|x, y\|_{\mathfrak{C},v}$, ultrametric inequality for      104 224
Cantor set      347
Cantor, D.      2 6 10 16 50 220 226 320 328 332 355 369 381
Capacity, (Archimedean case), capacity of $\partial D_{\zeta}$, $E^*_{\zeta}$      146
Capacity, (Archimedean case), definition, $\gamma_{\zeta}(E)$ for compact sets      136
Capacity, (Archimedean case), definition, $\gamma_{\zeta}(E)$ for compact sets, for arbitrary sets      137
Capacity, (Archimedean case), equality with $d_{\zeta}(E)$      151
Capacity, (Archimedean case), examples      338347
Capacity, (Archimedean case), limits of open sets      149
Capacity, (Archimedean case), sets of capacity 0      133 134 137 137
Capacity, (Archimedean case), table of capacities      348351
Capacity, (Archimedean case), unchanged by sets of capacity, capacity 0      149
Capacity, (Nonarchimedean case) for countable sets      200
Capacity, (Nonarchimedean case), algorithm for computing capacity of a union of sets      353355
Capacity, (Nonarchimedean case), countable unions      193
Capacity, (Nonarchimedean case), definition, $\gamma_{\zeta}(E)$ for compact sets      190
Capacity, (Nonarchimedean case), definition, $\gamma_{\zeta}(F)$ for general sets      259
Capacity, (Nonarchimedean case), definition, $\gamma_{\zeta}(U)$ for PL-domains      259
Capacity, (Nonarchimedean case), dependence on ground field      352
Capacity, (Nonarchimedean case), equality with $d_{\zeta}(E)$      204
Capacity, (Nonarchimedean case), examples      211219 352365
Capacity, (Nonarchimedean case), examples for Elliptic curves      360365
Capacity, (Nonarchimedean case), inner capacity $\gamma_{\zeta}(F)$      15 192 259
Capacity, (Nonarchimedean case), limits of open sets      200
Capacity, (Nonarchimedean case), outer capacity $\gamma_{\zeta}(E)$      15 259
Capacity, (Nonarchimedean case), unchanged by capacity 0 sets      199
Capacity, (Nonarchimedean case), unions of compact sets      201
Capacity, , (Global capacity)      3 10 171 163
Capacity, , (Global capacity) for elliptic curves      5 370
Capacity, , (Global capacity), algorithm for computing $val(\Gamma)$      366367
Capacity, , (Global capacity), base extension      333
Capacity, , (Global capacity), definition of $\gamma(\mathbb{F})$      328
Capacity, , (Global capacity), monotonicity properties      331332
Capacity, , (Global capacity), pullbacks      333
Capacity, , (Global capacity), separation inequality on $P^1$      333
Carrier $E^*_{\zeta}$      140 146 195
Cauchy estimates      34 35 36
Centers $x_i\in \mathfrak{X}$      322
Chebyshev constants      7 150 203
Chinbnrg, T.      19 79 103
Choqust, G.      265
Chordal distance      23 25 26
Chordal distance as Arakelov Greens function      81
Circuit model      110
Codisc      22
Conductor potential $u_E(z,\zeta)$, (Archimedean case), constant on E      143
Conductor potential $u_E(z,\zeta)$, (Archimedean case), continuity properties      142
Conductor potential $u_E(z,\zeta)$, (Archimedean case), definition      137
Conductor potential $u_E(z,\zeta)$, (Archimedean case), uniqueness      145
Conductor potential $u_E(z,\zeta)$, (Nonarchimsdsan case), constant on E      195
Conductor potential $u_E(z,\zeta)$, (Nonarchimsdsan case), definition      191
Conductor potential $u_E(z,\zeta)$, (Nonarchimsdsan case), examples      211219
Conductor potential $u_E(z,\zeta)$, (Nonarchimsdsan case), examples for finite unions      216
Conductor potential $u_E(z,\zeta)$, (Nonarchimsdsan case), examples for pullbacks      214
Conductor potential $u_E(z,\zeta)$, (Nonarchimsdsan case), examples for rings of integers      212
Conductor potential $u_E(z,\zeta)$, (Nonarchimsdsan case), less than $V_{\zeta}(E)$ off E      197
Conductor potential $u_E(z,\zeta)$, (Nonarchimsdsan case), pathological examples      217
Conductor potential $u_E(z,\zeta)$, (Nonarchimsdsan case), uniqueness      211
Continuum      156
Convex hull      174
Coordinate patches      133
Currents      86
deg(f)      21
Degree sequence      381
Deligne Mumford Theorem      101 118
Dictionary between circuits, harmonic functions on graphs      110
Differentials of 1st, 3rd kinds      64
Dirichlet Minkowski Unit Theorem      395
Disc with respect to $\|x, y\|_{\mathfrak{C},v}$      222
Disc, boundary of      30
Disc, conventions      22 356
Disc, isometrically parametrizable      30
Disc, open, closed      30
Divisor function      113
Dwork, B.      17 39 40
Electric circuit analogy      110
Electrostatic analogy for capacity      6 8
Elliptic curve      5 361 370
Energy integral      136
Equilibrium distribution      7
Equilibrium distribution, (Archimedean case), definition      137
Equilibrium distribution, (Archimedean case), examples of      163
Equilibrium distribution, (Archimedean case), formula in terms of $G(z,\omega,E)$      162
Equilibrium distribution, (Archimedean case), uniqueness      145
Equilibrium distribution, (Nonarchimedean case), definition      190
Equilibrium distribution, (Nonarchimedean case), examples      211219
Equilibrium distribution, (Nonarchimedean case), examples for finite unions of sets      216
Equilibrium distribution, (Nonarchimedean case), examples for pullbacks      214
Equilibrium distribution, (Nonarchimedean case), examples for rings of integers      212213
Equilibrium distribution, (Nonarchimedean case), pathological examples      217
Equilibrium distribution, (Nonarchimedean case), uniqueness      211
Evans function      208
Exponential map      49
Fekete points      152
Fekete Szego Theorem for algebraic curves      4 414 415
Fekete Szego Theorem in classical case      1 2 373 376
Fekete Szego Theorem when $\gamma(\mathbb{F},\mathfrak{X}) = 1$      416
Fekete Szego Theorem, archimedean input      171 173
Fekete Szego Theorem, inner vs outer capacity appropriate for      418420
Fekete Szego Theorem, nonarchimedean input      316 318 319
Fekete, M. and Szego.G.      1 2 3 4 6 9 15 373
Fine cover      187
Fine subcover      187
Frobeniuss Theorem      328329
Frostmans Theorem      140 195 284
Frssnsl, J.      220
Fubini Study metric      23 26
Fulton, V.      117
Function suitable for defining $[x, y]_{\zeta}$      59
Fundamental group      63
Fundamental Theorem of Game Theory      327 335
Generalized disc      222
Generic value of $|f(z)|_v$      223
Generic value of $|f(z)|_v$, $g_a(r, f)$      224
Generic value of $|f(z)|_v$, $g_{\matcal{R}}(\chi, f)$      224
Geodesic      26
Gillet, H.      18
Global mapping functions      381
Global mapping functions, when $\gamma(\mathbb{F},\mathfrak{X}) < 1$      384393
Global mapping functions, when $\gamma(\mathbb{F},\mathfrak{X}) > 1$      395413
Good reduction      91 322 323 383 411
Greens function $G(z, \zeta,E)$      8 10 14 155 277
Greens function $G(z, \zeta,E)$, (Archimedean case), approximatable by $G(z,\zeta,E)$ of good sets      165
Greens function $G(z, \zeta,E)$, (Archimedean case), approximatable by (1/N)log|f(z)|      168
Greens function $G(z, \zeta,E)$, (Archimedean case), basic properties      156
Greens function $G(z, \zeta,E)$, (Archimedean case), behavior under limits      158159
Greens function $G(z, \zeta,E)$, (Archimedean case), continuity in two variables      161
Greens function $G(z, \zeta,E)$, (Archimedean case), definition of      155
Greens function $G(z, \zeta,E)$, (Archimedean case), functoriality      163 164
Greens function $G(z, \zeta,E)$, (Archimedean case), monotonicity      157
Greens function $G(z, \zeta,E)$, (Archimedean case), positivity off E      156
Greens function $G(z, \zeta,E)$, (Archimedean case), symmetry      160
Greens function $G(z, \zeta,E)$, (Nonarchimedean case), $\underline{G}(z,\zeta;F) <\bar{G}(z,\zeta;F)$      278 282
Greens function $G(z, \zeta,E)$, (Nonarchimedean case), $\underline{G}(z,\zeta;F) = \bar{G}(z,\zeta;F)$ off F, for A.C. sets      283 291 293
Greens function $G(z, \zeta,E)$, (Nonarchimedean case), behavior under pullbacks      305
Greens function $G(z, \zeta,E)$, (Nonarchimedean case), continuity of $G(z,\zeta;F)$      301
Greens function $G(z, \zeta,E)$, (Nonarchimedean case), definition: $\underline{G}(z,\zeta;F) = G(z,\zeta;F)$ for A.C. sets      297
Greens function $G(z, \zeta,E)$, (Nonarchimedean case), dependence on ground field      35
Greens function $G(z, \zeta,E)$, (Nonarchimedean case), examples of non-A.C, sets, where upper, lower Greens functions differ      282283
Greens function $G(z, \zeta,E)$, (Nonarchimedean case), examples with $\underline{G}(z,\zeta;F) \neq \bar{G}(z,\zeta;F)$ on F      295
Greens function $G(z, \zeta,E)$, (Nonarchimedean case), Galois stability      310
Greens function $G(z, \zeta,E)$, (Nonarchimedean case), lower: $\underline{G}(z,\zeta;F)$      277 282
Greens function $G(z, \zeta,E)$, (Nonarchimedean case), monotonicity in F      297
Greens function $G(z, \zeta,E)$, (Nonarchimedean case), sequence Greens functions $\underline{G}(z,\zeta;\{E_j\})$, $\bar{G}(z,\zeta;\{D_k\})$      283
Greens function $G(z, \zeta,E)$, (Nonarchimedean case), symmetry of $G(z,\zeta;F)$      299
Greens function $G(z, \zeta,E)$, (Nonarchimedean case), the zero set $S_0(F,\zeta)$      292
Greens function $G(z, \zeta,E)$, (Nonarchimedean case), upper: $\bar{G}(z,\zeta;F)$      277 282
Greens function $G(z, \zeta,E)$, (Nonarchimedean case), upper: $\bar{G}_{\mu}(z,\zeta)$      208
Greens function $G(z, \zeta,E)$, (Nonarchimedean case), well defined      278
Greens identity      67 87 160 161
Greens matrix, $\Gamma(\mathbb{F},\mathfrak{X})$ well-defined      327
Greens matrix, $\Gamma_v=0$ for almost all v      324
Greens matrix, global Greens matrix $\Gamma(\mathbb{F},\chi)$      1 325 326
Greens matrix, Gross, B.      12 19 63 74 77 83 89 106
Greens matrix, local Greens matrix $\Gamma_v$      11 324 326
Greens matrix, reducible vs irreducible      328 369
Grothendieck, A.      99
Gunning, R.      63 64 65
Haar measure      23
Hadamard Quotient Theorem      18
Half-disc      340
Harbatsr, D.      19 27
Harmonic function      63 73
Harnacks principle      70
Hartogs, F.      65
High order coefficients      374 378 389 408
Hilberts 10th problem for $\tilde{\mathcal{O}}$      6
Hilberts Lemniscate Theorem      168
Hills, E.      168 374
Hole (of codisc)      223 225
Horizontal divisor      98
Hriljac, P.      106
Hyperbolic polygon group      83
Independent variability of leading coefficients      173
Inner capacity      192
Inner capacity for a PL-domain      258
Intersection pairing      98
Intersection pairing for divisors of functions      100
Intersection pairing for horizontal divisors      99
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